Family

The formal term used for a collection of objects. It is denoted {a_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline1.svg" style="height:24px; width:46px" /> (but other kinds of brackets can be used as well), where  is a nonempty set called the index set, and  is called the term of index  of the family.

A family with index set  is called a sequence.

The union and the intersection of a family of sets {A_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline6.svg" style="height:24px; width:49px" /> are denoted

(1)

respectively.

If all terms  belong to an additive monoid, one can consider the sum

(2)

provided the number of nonzero terms is finite, i.e., the so-called support of the family

{i in I|a_i!=0} " src="https://mathworld.wolfram.com/images/equations/Family/NumberedEquation3.svg" style="height:22px; width:94px" />

(3)

is a finite set. A similar argument applies to multiplicative monoids, and to the product

(4)

up to replacement of the zero element with the identity element 1.

According to its formal definition (Bourbaki 1970), if the terms  belong to the set , the family {a_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline10.svg" style="height:24px; width:46px" /> is a map , where  for all .

Every set  gives rise to a family

(5)

from which the original set can be recovered as the range of . Accordingly, every family  also gives rise to a set

{a_i|i in I}, " src="https://mathworld.wolfram.com/images/equations/Family/NumberedEquation6.svg" style="height:22px; width:106px" />

(6)

from which, however, the original family in general cannot be recovered.

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REFERENCES

Bourbaki, N. Eléments de Mathématiques. Théorie des Ensembles. Paris, France: Hermann, p. ER11, 1970.

مواضيع ذات صلة

Subset

Set Class

Set

Second Category

Proper Subset

Power Set

Matroid

Inductive Set

Independent Vertex Set

Independent Set

Graphoid

G_delta Set